Vectors

(8) Polar vectors : These have starting point or point of

VECTOR ALGEBRA application . Example displacement and force etc.
(9) Axial Vectors : These represent rotational effects and are
always along the axis of rotation in accordance with right hand screw
Introduction of Vector rule. Angular velocity, torque and angular momentum, etc., are
example of physical quantities of this type.
Physical quantities having magnitude, direction and obeying
laws of vector algebra are called vectors. Axial vector Axis of rotation
Example : Displacement, velocity, acceleration, momentum,
force, impulse, weight, thrust, torque, angular momentum, angular
velocity etc.
If a physical quantity has magnitude and direction both, then it Anticlock wise rotation Clock wise rotation
does not always imply that it is a vector. For it to be a vector the third
Axis of rotation Axial vector
condition of obeying laws of vector algebra has to be satisfied.
Example : The physical quantity current has both magnitude
(10) Coplanar vector : Three (or more) vectors are called
and direction but is still a scalar as it disobeys the laws of vector
coplanar vector if they lie in the same plane. Two (free) vectors are
algebra.
always coplanar.
Types of Vector
Triangle Law of Vector Addition of Two Vectors
(1) Equal vectors : Two vectors A and B are said to be equal when If two non zero vectors are represented by the two sides of a
they have equal magnitudes and same direction. triangle taken in same order B
(2) Parallel vector : Two vectors A and B are said to be then the resultant is given by
the closing side of triangle in R  AB
parallel when
(i) Both have same direction. opposite order. i.e. R  A  B B
(ii) One vector is scalar (positive) non-zero multiple of another  OB  OA  AB O A
vector. A

(3) Anti-parallel vectors : Two vectors A and B are said to (1) Magnitude of resultant vector
be anti-parallel when
AN
(i) Both have opposite direction. In  ABN , cos   AN  B cos
B
(ii) One vector is scalar non-zero negative multiple of another
BN
vector. sin   BN  B sin
(4) Collinear vectors : When the vectors under consideration B
can share the same support or have a common support then the In OBN , we have OB 2  ON 2  BN 2
considered vectors are collinear. B

(5) Zero vector (0 ) : A vector having zero magnitude and R

B B sin
arbitrary direction (not known to us) is a zero vector.
 
(6) Unit vector : A vector divided by its magnitude is a unit vector.
O A A N
Unit vector for A is Aˆ (read as A cap or A hat).
B cos
ˆ  A  A AA
Since, A ˆ .
A  R 2  ( A  B cos  )2  (B sin )2
Thus, we can say that unit vector gives us the direction.  R 2  A 2  B 2 cos 2   2 AB cos   B 2 sin2 
(7) Orthogonal unit vectors ˆi , ˆj  R 2  A 2  B 2 (cos 2   sin2  )  2 AB cos 
y
and kˆ are called orthogonal unit vectors.  R 2  A 2  B 2  2 AB cos 
These vectors must form a Right Handed ˆj
Triad (It is a coordinate system such that  R A 2  B 2  2 AB cos
when we Curl the fingers of right hand k̂ x
from x to y then we must get the direction î (2) Direction of resultant vectors : If  is angle between A
of z along thumb). The z
and B, then
ˆi  x , ˆj  y , kˆ  z
x y z | A  B|  A 2  B 2  2 AB cos

 x  xˆi , y  yˆj , z  zkˆ If R makes an angle  with A, then in OBN ,

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 Vectors
BN BN  Resultant of three co-planar vectors may or may not
tan   
ON OA  AN be zero

B sin  Resultant of three non co- planar vectors can not be zero.
tan  
A  B cos Subtraction of vectors
Parallelogram Law of Vector Addition
If two non zero vectors are represented by the two adjacent Since, A  B  A  ( B) and
sides of a parallelogram then the resultant is given by the diagonal of
the parallelogram passing through the point of intersection of the two | A  B |  A 2  B 2  2 AB cos
vectors.
 | A  B|  A 2  B 2  2 AB cos (180 o   )
(1) Magnitude
Since, R 2  ON 2  CN 2 Since, cos (180   )   cos
 R  (OA  AN )  CN
2 2 2
 | A  B |  A 2  B 2  2 AB cos 
 R  A  B  2 AB cos 
2 2 2

R sum  A  B
 R | R | | A  B |  A 2  B 2  2 AB cos 
B C B

 1
R  AB
B B sin 2 A
B 180 – 
 B
 
O A N
A B cos
R diff  A  ( B)

Special cases : R  A  B when  = 0o

R  A  B when  = 180o B sin
tan  1 
A  B cos
R A 2  B 2 when  = 90o
B sin(180   )
(2) Direction and tan  2 
A  B cos (180   )
CN B sin
tan    But sin(180   )  sin and cos(180   )   cos
ON A  B cos

Polygon Law of Vector Addition B sin

 tan  2 
A  B cos
If a number of non zero vectors are represented by the (n –
1) sides of an n-sided polygon then the resultant is given by the closing Resolution of Vector Into Components
side or the nth side of the polygon taken in opposite order. So,
Consider a vector R in X-Y plane
R  ABCD E Y
as shown in fig. If we draw orthogonal
OA  AB  BC  CD  DE  OE vectors R x and R y along x and y axes
respectively, by law of vector addition, Ry R
D D C
R  Rx  R y 
X
E C Rx
Now as for any vector A  A nˆ so,
E B R x  ˆi R x and R y  ˆjR y

R B so R  ˆi R x  ˆjRy …(i)

O A But from figure R x  R cos  …(ii)

A
and R y  R sin …(iii)
Note :  Resultant of two unequal vectors can not be zero.

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 Vectors
Since R and  are usually known, Equation (ii) and (iii) give the Scalar Product of Two Vectors
magnitude of the components of R along x and y-axes respectively. (1) Definition : The scalar product (or dot product) of two
Here it is worthy to note once a vector is resolved into its vectors is defined as the product of the magnitude of two vectors with
components, the components themselves can be used to specify the cosine of angle between them.
vector as
Thus if there are two vectors A and B having angle 
(1) The magnitude of the vector R is obtained by squaring and
between them, then their scalar product written as A . B is defined as
adding equation (ii) and (iii), i.e.
A . B  AB cos
R  R x2  R y2 B
(2) Properties : (i) It
(2) The direction of the vector R is obtained by dividing is always a scalar which is 
equation (iii) by (ii), i.e. positive if angle between
the vectors is acute (i.e., <
tan   (Ry / R x ) or   tan 1 (Ry / R x ) 90°) and negative if angle A
between them is obtuse (i.e.
Rectangular Components of 3-D Vector 90°< < 180°).

R  R x  R y  R z q or R  R x ˆi  R y ˆj  R z kˆ (ii) It is commutative, i.e. A . B  B . A

Y
(iii) It is distributive, i.e. A . (B  C)  A . B  A . C

(iv) As by definition A . B  AB cos 

R
Ry
Rx  A.B 
The angle between the vectors   cos 1  
 AB 
Rz
X
(v) Scalar product of two vectors will be maximum when
Z
cos  max  1, i.e.   0 o , i.e., vectors are parallel

( A . B)max  AB
If R makes an angle  with x axis,  with y axis and  with z
axis, then (vi) Scalar product of two vectors will be minimum when
| cos  |  min  0, i.e.   90 o
Rx Rx
 cos    l
R R x2  R y2  R z2 ( A . B)min  0

i.e. if the scalar product of two nonzero vectors vanishes the

Ry Ry
 cos    m vectors are orthogonal.
R R x2  R y2  R z2
(vii) The scalar product of a vector by itself is termed as self
Rz Rz dot product and is given by ( A)2  A . A  AA cos   A 2
 cos    n
R R x2  R y2  R z2
i.e. A  A. A
Where l, m, n are called Direction Cosines of the vector R and (viii) In case of unit vector n̂
R x2  R y2  R z2 nˆ . nˆ  1  1  cos 0  1 so nˆ . nˆ  ˆi .ˆi  ˆj . ˆj  kˆ . kˆ  1
l 2  m 2  n 2  cos 2   cos 2   cos 2   1
R x2  R y2  R z2
(ix) In case of orthogonal unit vectors ˆi , ˆj and kˆ ,
Note :  When a point P have coordinate (x, y, z) ˆi . ˆj  ˆj . kˆ  kˆ . ˆi  1  1 cos 90  0
then its position vector OP  xˆi  yˆj  zkˆ (x) In terms of components
 When a particle moves from point (x1, y1, z1) to (x2,
A . B  (iAx  jAy  k Az ). (iBx  jBy  k Bz )  [ Ax Bx  Ay By  AZ Bz ]
y2, z2) then its displacement vector
 (3) Example : (i) Work W : In physics for constant force work
r  (x 2  x 1 )ˆi  (y 2  y1 )ˆj  (z 2  z1 )kˆ is defined as, W  Fs cos  …(i)

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 Vectors
But by definition of scalar product of two vectors, The direction of A  B, i.e. C is perpendicular to the plane
F. s  Fs cos  …(ii)
containing vectors A and B and in the sense of advance of a right
So from eq (i) and (ii) W  F.s i.e. work is the scalar product
n
handed screw rotated from A (first vector) to B (second vector)
of force with displacement. through the smaller angle between them. Thus, if a right handed screw
(ii) Power P : whose axis is perpendicular to the plane framed by A and B is

dW ds rotated from A to B through the smaller angle between them, then

As W  F . s or  F. [As F is constant] the direction of advancement of the screw gives the direction of
dt dt
A  B i.e. C
or P  F . v i.e., power is the scalar product of force with
 dW ds 
velocity.  As  P and  v  (2) Properties
 dt dt  ds 
B (i) Vector product of any two vectors is always a vector
(iii) Magnetic Flux  : perpendicular to the plane containing these two vectors, i.e.,
Magnetic flux through an area 
orthogonal to both the vectors A and B, though the vectors A and
is given by d  B ds cos …(i)
B may or may not be orthogonal.
O
But by definition of scalar (ii) Vector product of two vectors is not commutative, i.e.,
product B . d s  Bds cos ...(ii) A  B  B  A [but   B  A]
n
So from eq (i) and (ii) we have Here it is worthy to note that

d  B . d s or    B.ds | A  B | | B  A |  AB sin

(iv) Potential energy of a dipole U : If an electric dipole of i.e. in case of vector A  B and B  A magnitudes are equal
moment p is situated in an electric field E or a magnetic dipole of but directions are opposite.

moment M in a field of induction B, the potential energy of the (iii) The vector product is distributive when the order of the
vectors is strictly maintained, i.e.
dipole is given by :
A  (B  C)  A  B  A  C
U E   p . E and U B   M . B
(iv) The vector product of two vectors will be maximum when
Vector Product of Two Vectors
sin  max  1, i.e.,   90 o
(1) Definition : The vector product or cross product of two
vectors is defined as a vector having a magnitude equal to the product [ A  B]max  AB nˆ
of the magnitudes of two vectors with the sine of angle between them,
and direction perpendicular to the plane containing the two vectors in i.e. vector product is maximum if the vectors are orthogonal.
accordance with right hand screw rule. (v) The vector product of two non- zero vectors will be minimum

C  AB when | sin |  minimum = 0, i.e.,   0 o or 180 o

Thus, if A and B are two vectors, then their vector product [ A  B]min  0
written as A  B is a vector C defined by i.e. if the vector product of two non-zero vectors vanishes, the
vectors are collinear.
C  A  B  AB sin nˆ
(vi) The self cross product, i.e., product of a vector by itself
vanishes, i.e., is null vector A  A  AA sin 0 o nˆ  0

(vii) In case of unit vector nˆ  nˆ  0 so that

ˆi  ˆi  ˆj  ˆj  kˆ  kˆ  0

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 Vectors
(viii) In case of orthogonal unit vectors, ˆi , ˆj, kˆ in accordance i.e. for any triangle the ratio of the sine of the angle containing
the side to the length of the side is a constant.
with right hand screw rule :
For a triangle whose three sides are in the same order we
ˆj ˆj establish the Lami's theorem in the following manner. For the triangle
shown

k̂ a  b  c  0 [All three sides are taken in order] …(i)

î
î  a  b  c …(ii)

k̂ Pre-multiplying both sides by a

a  (a  b)  a  c  0  a  b  a  c
ˆi  ˆj  kˆ , ˆj  kˆ  ˆi and kˆ  ˆi  ˆj

And as cross product is not commutative,  ab  ca …(iii)

ˆj  ˆi  kˆ , kˆ  ˆj  ˆi and ˆi  kˆ  ˆj Pre-multiplying both sides of (ii) by b

(x) In terms of components b  (a  b)   b  c  b  a  b  b  b  c

ˆi ˆj kˆ   a  b  b  c  a  b  b  c …(iv)
A  B  Ax Ay Az
Bx By Bz From (iii) and (iv), we get a  b  b  c  c  a

Taking magnitude, we get | a  b | | b  c | | c  a |

 ˆi ( Ay Bz  Az By )  ˆj( A z B x  A x B z )  kˆ ( A x B y  A y B x )
 ab sin(180   )  bc sin(180   )  ca sin(180   )
(3) Example : Since vector product of two vectors is a vector,
vector physical quantities (particularly representing rotational effects)  ab sin  bc sin  ca sin 
like torque, angular momentum, velocity and force on a moving charge
in a magnetic field and can be expressed as the vector product of two Dividing through out by abc, we have
vectors. It is well – established in physics that :
sin sin  sin
  
(i) Torque   r  F a b c

(ii) Angular momentum L  r  p Relative Velocity

(1) Introduction : When we consider the motion of a particle,
(iii) Velocity v    r
we assume a fixed point relative to which the given particle is in
(iv) Force on a charged particle q moving with velocity v in a motion. For example, if we say that water is flowing or wind is
blowing or a person is running with a speed v, we mean that these all
magnetic field B is given by F  q(v  B) are relative to the earth (which we have assumed to be fixed).
(v) Torque on a dipole in a field  E  p  E and  B  M  B Y
Y P
Lami's Theorem
r
In any  A B C with sides a, b, c r
PS '
PS

sin sin  sin X

S
  rS ' S
a b c
S X
180 – 

 Now to find the velocity of a moving object relative to another

moving object, consider a particle P whose position relative to frame S
c b  
is rPS while relative to S  is rPS  .
  180 –  
If the position of frames S  relative to S at any time is r S S

180 –  a   
then from figure, rPS  rPS   rS S

Differentiating this equation with respect to time

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 Vectors
  
drPS drPS  drS S v RM  v R2  v M
2
 
dt dt dt
  
direction   tan 1 (v M / v R ) with the vertical as shown in fig.
 
or v PS  v PS   v S S [as v  d r /dt ]
  
or v PS   v PS  v S S
 – vM 
(2) General Formula : The relative velocity of a particle P1 vR vR

  vR 
moving with velocity v1 with respect to another particle P2 moving vM vM
   
with velocity v 2 is given by, v r1 2 = v1 – v 2

v1
(5) Relative velocity of swimmer : If a man can swim relative
v2 
P2 to water with velocity v and water is flowing relative to ground with
 

P1
velocity v R velocity of man relative to ground v M will be given by:

(i) If both the particles are moving in the same direction then :
r12  1 –  2      
v  v M  v R , i.e., v M  v  v R
(ii) If the two particles are moving in the opposite direction,
then : So if the swimming is in the direction of flow of water,
r12  1   2 vM  v  vR
(iii) If the two particles are moving in the mutually
And if the swimming is opposite to the flow of water,
perpendicular directions, then:
vM  v  vR
r12  12   22
(6) Crossing the river : Suppose, the river is flowing with
 
  velocity  r . A man can swim in still water with velocity  m . He is
(iv) If the angle between 1 and 2 be , then
standing on one bank of the river and wants to cross the river, two

r12  12   22 – 21 2 cos  
1/ 2
. cases arise.

(3) Relative velocity of satellite : If a satellite is moving in (i) To cross the river over shortest distance : That is to cross the
 river straight, the man should swim making angle  with the upstream
equatorial plane with velocity v s and a point on the surface of earth as shown.
 
with v e relative to the centre of earth, the velocity of satellite relative A vr B
to the surface of earth 

 
   vr
vse  v s  v e w  v
vm

So if the satellite moves form west to east (in the direction of

rotation of earth on its axis) its velocity relative to earth's surface will
Upstream O Downstream
be v se  v s  ve

And if the satellite moves from east to west, i.e., opposite to the Here OAB is the triangle of vectors, in which
motion of earth, v se  v s  (ve )  v s  ve      
OA  vm , AB  r . Their resultant is given by OB   . The direction
(4) Relative velocity of rain : If rain is falling vertically with a of swimming makes angle  with upstream. From the triangle OBA,
 
we find,
velocity v R and an observer is moving horizontally with speed v M
r 
the velocity of rain relative to observer will be v RM  v R  v M
   cos   Also sin  r
m m
which by law of vector addition has magnitude

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 Vectors
Where  is the angle made by the direction of swimming with  Vectors are added and subtracted geometrically.
the shortest distance (OB) across the river.
 Division of vectors is not allowed as directions cannot be
Time taken to cross the river : If w be the width of the river, divided.
then time taken to cross the river will be given by
 Unit vector gives the direction of vector.
w w
t1    Magnitude of unit vector is 1.
 m2 – r2
 Unit vector has no unit. For example, velocity of an object is 5
(ii) To cross the river in shortest possible time : The man should ms–1 due East.
swim perpendicular to the bank. 
i.e. v  5ms 1 due east.
The time taken to cross the river will be: 
v 5 ms 1 (East)
vˆ     East
t2 
w | v| 5 ms 1
m
So unit vector v̂ has no unit as East is not a physical quantity.

A vr B
 Unit vector has no dimensions.

 ˆi . ˆi  ˆj . ˆj  kˆ . kˆ  1
 
w vr 
vm
 ˆi  ˆi  ˆj  ˆj  kˆ  kˆ  0

 ˆi  ˆj  kˆ , ˆj  kˆ  ˆi , kˆ  ˆi  ˆj
Upstream O Downstream
 ˆi . ˆj  ˆj . kˆ  kˆ . ˆi  0

In this case, the man will touch the opposite bank at a distance          
 A  A  0 . Also A  A  0 But A  A  A  A
AB down stream. This distance will be given by:      
Because A  A  A and A  A is collinear with A
w r
AB  r t 2 r or AB  w
m m  Multiplication of a vector with –1 reverses its direction.
 
If A  B , then A = B and A ˆ B ˆ .
POINTS TO REMEMBER   
If A  B  0 , then A = B but A ˆ  B
ˆ .

 All physical quantities having direction are not vectors. For  Minimum number of collinear vectors whose resultant can be
example, the electric current possesses direction but it is a zero is two.
scalar quantity because it can not be added or multiplied
according to the rules of vector algebra.  Minimum number of coplaner vectors whose resultant is zero
is three.
 A vector can have only two rectangular components in plane
and only three rectangular components in space.  Minimum number of non coplaner vectors whose resultant is
zero is four.
 A vector can have any number, even infinite components.  
(minimum 2 components)  Two vectors are perpendicular to each other if A.B  0 .
 
 Following quantities are neither vectors nor scalars : Relative  Two vectors are parallel to each other if A  B  0.
density, density, viscosity, frequency, pressure, stress, strain,
modulus of elasticity, poisson’s ratio, moment of inertia,  Displacement, velocity, linear momentum and force are polar
specific heat, latent heat, spring constant loudness, resistance, vectors.
conductance, reactance, impedance, permittivity, dielectric
 Angular velocity, angular acceleration, torque and angular
constant, permeability, susceptibility, refractive index, focal
momentum are axial vectors.
length, power of lens, Boltzman constant, Stefan’s constant,
Gas constant, Gravitational constant, Rydberg constant,  Division with a vector is not defined because it is not possible
Planck’s constant etc. to divide with a direction.
 Distance covered is a scalar quantity.  Distance covered is always positive quantity.
 The displacement is a vector quantity.  The components of a vectors can have magnitude than that of
 Scalars are added, subtracted or divided algebraically. the vector itself.

 The rectangular components cannot have magnitude greater

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 Vectors
 
than that of the vector itself.  Projection of a vector B in the direction of vector A
 
 When we multiply a vector with 0 the product becomes a null A. B
 
vector. | A|
  
 The resultant of two vectors of unequal magnitude can never  If vectors A, B and C are represented by three sides ab, bc
be a null vector.
and ca respectively taken in a order, then
  
 Three vectors not lying in a plane can never add up to give a | A| | B| | C|
null vector.  
ab bc ca
 A quantity having magnitude and direction is not necessarily a  The vectors ˆi  ˆj  kˆ is equally inclined to the coordinate axes
vector. For example, time and electric current. These quantities at an angle of 54.74 degrees.
have magnitude and direction but they are scalar. This is      
because they do not obey the laws of vector addition.  If A  B  C , then A . B  C  0 .
     
 A physical quantity which has different values in different  If A . B  C  0 , then A . B and C are coplanar.
 
directions is called a tensor. For example : Moment of inertia  If angle between A and B is 45°,
has different values in different directions. Hence moment of    
then A . B | A  B |
inertia is a tensor. Other examples of tensor are refractive
    
index, stress, strain, density etc.  If A1  A2  A3  ......  An  0 and A1  A2  A3  ......  An
 The magnitude of rectangular components of a vector is then the adjacent vector are inclined to each other at angle
always less than the magnitude of the vector 2 / n .
 
 If A  B , then A x  B x , Ay  By and Az  Bz .    
 If A  B  C and A 2  B 2  C 2 , then the angle between A
          
 If A  B  C . Or if A  B  C  0 , then A, B and C lie in and B is 90°. Also A, B and C can have the following values.
one plane. (i) A = 3, B = 4, C = 5
      (ii) A = 5, B = 12, C = 13
 If A  B  C , then C is perpendicular to A as well as B . (iii) A = 8, B = 15, C = 17.
     
 If | A  B | | A  B | , then angle between A and B is 90°.
 Resultant of two vectors will be maximum when  = 0° i.e.
vectors are parallel.
Rmax  P 2  Q 2  2 PQ cos 0 | P  Q |
 Resultant of two vectors will be minimum when  = 180° i.e.
vectors are anti-parallel.
Rmin  P 2  Q 2  2 PQ cos 180 | P  Q |
Thus, minimum value of the resultant of two vectors is equal to the
difference of their magnitude.
 Thus, maximum value of the resultant of two vectors is equal
to the sum of their magnitude.
When the magnitudes of two vectors are unequal, then
 
Rmin  P  Q  0 [| P | | Q |]
 
Thus, two vectors P and Q having different magnitudes can never
be combined to give zero resultant. From here, we conclude that the
minimum number of vectors of unequal magnitude whose resultant
can be zero is three. On the other hand, the minimum number of
vectors of equal magnitude whose resultant can be zero is two.
 
 Angle between two vectors A and B is given by
 
A.B
cos   
| A| | B|
 
 Projection of a vector A in the direction of vector B
 
A. B
 
| B|

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